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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">s</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>;<br/>



<b>set </b><font color="Maroon" title="c3">X</font> =  <a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">s</font>;<br/>
<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">s</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c6">P</font>, <font color="Maroon" title="c7">Q</font> be   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">s</font>;<br/>



<b>assume </b><a NAME="E1:58"/><i><font color="Green" title="E37">A1</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">s</font></span>)</span>,<font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">s</font></span>)</span>,<font color="Maroon" title="c7">Q</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">P1</font>, <font color="Maroon" title="c9">Q1</font> being   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">s</font><b> such that </b><br/><a NAME="E3:58"/><i><font color="Green" title="E38">A2</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">s</font></span>)</span>,<font color="Maroon" title="c8">P1</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">s</font></span>)</span>,<font color="Maroon" title="c9">Q1</font> )
 <b>and </b><br/><a NAME="E4:58"/><i><font color="Green" title="E39">A3</font></i>: 
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">s</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">P1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c9">Q1</font></span>)</span>
 <b>by </b><i><a class="ref" href="topalg_1.html#D3" target="_self" title="TOPALG_1:def.3">Def3</a></i>;<br/>
<a NAME="E5:58"/>
( <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c8">P1</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  &amp; <font color="Maroon" title="c7">Q</font>,<font color="Maroon" title="c9">Q1</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  )
 
<b>by </b><i><a class="txt" href="topalg_1.html#E1:58"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="topalg_1.html#E3:58"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="topalg_1.html#T47" target="_self" title="TOPALG_1:th.47">Th47</a></i>;<br/>
<a NAME="E6:58"/><b>then </b>
<font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font>,<font color="Maroon" title="c8">P1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c9">Q1</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T83" title="BORSUK_6:th.83">BORSUK_6:83</a></i>;<br/>
<b>hence </b><a NAME="E7:58"/>
<font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_1.html#E4:58"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="topalg_1.html#T47" target="_self" title="TOPALG_1:th.47">Th47</a></i>;<br/>


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